Streamfunction-velocity computation of natural convection around heated bodies placed in a square enclosure

•Compact Ψ-v formulation on body fitted non-uniform grids without transformation.•First time, secondary corner vortices are shown to exist for the case of circular cylinder placed in a square enclosure.•Verification of the existence of vortices via pressure gradient observation.•Steady state results for Ra=107 for diamond cylinder placed in a square enclosure.•Comparison exercise of heated circular cylinder embedded in a square enclosure vis a vis its diamond cylinder counterpart. In the study of natural convection around embedded bodies in a square enclosure, most of the previous studies have used the primitive variable or the streamfunction-vorticity(ψ−ω) form of the Navier-Stokes (N-S) equations along with grid transformation for the computations. In the current study, we reconstruct a recently developed compact finite difference scheme for the biharmonic form of the N-S equations and combine it with a high order compact (HOC) scheme for the energy equation to compute the flow around heated circular and diamond cylinders inside a square enclosure. Our computed results on nonuniform grids without transformation are excellent match with available numerical results for both adiabatic and isothermal walls of the square. In the process, we have also been able to capture flow structures for certain configurations and range of parameters that were not reported earlier in the existing literature.